2-Selmer groups of hyperelliptic curves with marked points

Shankar, Ananth

doi:10.1090/tran/7546

$2$-Selmer groups of hyperelliptic curves with marked points
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by Ananth N. Shankar PDF

Trans. Amer. Math. Soc. 372 (2019), 267-304 Request permission

Abstract:

We consider the family of hyperelliptic curves over $\mathbb {Q}$ of fixed genus along with a marked rational Weierstrass point and a marked rational non-Weierstrass point. When these curves are ordered by height, we prove that the average Mordell–Weil rank of their Jacobians is bounded above by $5/2$, and that most such curves have only three rational points. We prove this by showing that the average rank of the $2$-Selmer groups is bounded above by $6$. We also consider another related family of curves and study the interplay between these two families. There is a family $\phi$ of isogenies between these two families, and we prove that the average size of the $\phi$-Selmer groups is exactly $2$.

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